Abstract and Applied Analysis
Volume 2012 (2012), Article ID 840919, 23 pages
http://dx.doi.org/10.1155/2012/840919
Research Article

The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

Ying Wang and YunXi Guo

College of Science, Sichuan University of Science and Engineering, Zigong 643000, China

Received 21 February 2012; Accepted 11 March 2012

Academic Editor: Shaoyong Lai

Copyright © 2012 Ying Wang and YunXi Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that ( 1 − 𝜕 2 𝑥 ) 𝑢 0 ∈ 𝑀 + ( 𝑅 ) , 𝑢 0 ∈ 𝐻 1 ( 𝑅 ) , and 𝑢 0 ∈ 𝐿 1 ( 𝑅 ) , the existence and uniqueness of the global weak solution to the equation are shown to be true.